The Neumann points of an eigenfunction f on a quantum (metric) graph are the interior zeros of \(f'\). The Neumann domains of f are the sub-graphs bounded by the Neumann points. Neumann points and Neumann domains are the counterparts of the well-studied nodal points and nodal domains. We prove bounds on the number of Neumann points and properties of the probability distribution of this number. Two basic properties of Neumann domains are presented: the wavelength capacity and the spectral position. We state and prove bounds on those as well as key features of their probability distributions. To rigorously investigate those probabilities, we establish the notion of random variables for quantum graphs. In particular, we provide conditions for considering spectral functions of quantum graphs as random variables with respect to the natural density on \({\mathbb {N}}\).

量子（度量）图上本征函数f的Neumann点是\（f'\）的内部零点。f的Neumann域是由Neumann点界定的子图。Neumann点和Neumann域是经过充分研究的节点和节点域的对应物。我们证明了诺伊曼点数的界和该数的概率分布的性质。介绍了Neumann域的两个基本属性：波长容量和光谱位置。我们陈述并证明其边界以及它们的概率分布的关键特征。为了严格研究这些概率，我们建立了量子图随机变量的概念。尤其是，我们提供了一些条件，可以将量子图的谱函数视为关于\（{{mathbb {N}} \）上的自然密度的随机变量。